The conditions of optimal stabilization of controlled dynamical systems described by nonlinear multiply connected systems of ordinary differential equations are considered. The properties of stabilizing control and form of integrand in criterion of quality transient are used taking into account that subsystems are asymptotically stable. The results are obtained for the case of a part of phase variables and for the case when right parts consist of homogeneous vectorfunctions. Conditions of optimal stabilization with respect to a part of variables are suggested. The algorithms of optimal stabilization of controlled multiply connected dynamical systems are designed.
Introduction The search of conditions of optimal stabilization and synthesis of corresponding stabilization algorithms is a significant problem in research of behavior of nonlinear controlled systems [Rumyantsev, 1970], [Krasovsky, 1966], [Rumyantsev, 1987], [Andreev, 1997]. Fundamental approach to optimal stabilization for systems of ordinary differential equations was developed by V.V. Rumyantsev with using of condition of minimization for a functional characterizing the quality of control. Solving of the problems of optimal stabilization for different types of multiply connected dynamic systems is based on the fundamental results [Rumyantsev, 1970] about optimal stabilization of nonlinear system of differential equations of perturbed motion with the additional forces. The functional is given in the form of a definite integral with the upper infinite limit. Integrand function of the functional is defined in the proof of the theorem, in this case the known Lyapunov function for the system of differential equations of perturbed motion without the control becomes the optimal Lyapunov function for the specified system under the action of additional forces.
Methods of optimal stabilization are considered in [Krasovsky, 1966], [Rumyantsev, 1987], [Andreev, 1997]. Critical cases are allocated and ways of a finding of stabilizing control in critical cases are developed [Galperin, 1963], [Hitrov, 1979]. The convenient way and the basis of general scheme of stabilization for multiply connected systems is two-level stabilization [Shil’yak, 1994]. Some methods for solving of the problem of optimal stabilization to respect to all variables and to a part of phase variables for multiply connected nonlinear controlled dynamic systems are given in [Shchennikova, 2006], [Druzhinina et al., 2011].
In this work we suggest the conditions and algorithms of optimal stabilization for controlled dynamic systems described by nonlinear multiply connected systems of ordinary differential equations. We consider the common case and the case when right-hand sides consists of homogeneous vector functions. The properties of stabilizing control and form of integrand in criteria of quality of transient are used taking into account that subsystems are asymptotically stable. Optimal control is synthesized at the level of the initial system. The results can be used in problems of control of motion of complicated spatial mechanisms, and also in problems of stabilization of motion of multiply connected systems of different types. 2
The Optimal Stabilization with Using of Homogeneous Vector Functions It is known that for nonlinear systems of differential equations of general form the conditions of theorems on asymptotic stability on the first nonlinear approach are hardly verified. However, in some cases is possible to search for fairly easily verifiable conditions under which we prove the asymptotic stability of the equilibrium of the first nonlinear approach. Nonlinear system with right-hand sides are homogeneous (generalized homogeneous) vector functions have been studied in [Kosov, 1997], [Alexandrov, 2004], which shows the theorems about asymptotic stability on the first nonlinear approach with the conditions which are easily verified. These results can be used in the solving of problems of optimal stabilization of nonlinear controlled multiply connected systems.
We consider multiply connected nonlinear controlled dynamic system dxs = Xs( s)(xs) + dt
q ∑ Rsj (t, x) + Bs(xs)us ≡ Φs(t, x, u).
j=1 Here xs ∈ Rns , x = (x1T , ..., xqT ), Xs( s)(xs) are homogeneous of order µs > 1 continuously differentiable vector functions, µs = ps/qs, ps and qs are odd numbers, us ∈ Rrs ,Rn1 × ... × Rnq = Rn, Rr1 × ... × Rrq = Rr, s = 1, q. Continious functions Rsj (t, x) are defined in domain It should be noted that we use the Euclidean norm of the vector in formula (2). It is accepted that the conditions Ω = {t, x : t ≥ t0, ||x|| < h, 0 < h = const}.
(1) (q) ||Rsj (t, x)|| ≤ csj ||x1|| sj . . . ||xq|| sj , csj ≥ 0, αs(ij) ≥ 0, dxs = Xs( s)(xs), dt
q ∑ αs(ij) > 1, i=1 s = 1, q, are hold. We assume that Φs(t, 0, 0) ≡ 0, s = 1, q, and that equilibrium states of systems are asymptotically stable. As a Lyapunov function for system (1) in this case we consider the function v(x) =
q ∑ vs(xs), s=1 where vs(xs) are Lyapunov functions for systems (3) satisfying the conditions: (i) vs(xs) and ws(xs) are positive definite functions; (ii) vs(xs) and ws(xs) are gomogeneously positive functions of order ms + 1 − µs and ms, where ms are enough large rational numbers with odd denominator and even numerator; (iii) functions vs(xs) are continuously differentiable and (∇vs)T Xs( s)(xs) = −ws(xs). (1) (2) (3)
The problem of optimal stabilization for the system (1) has a unique solution in closed form. Krasovsky function B[v; t, x, u] has a form
q B[v; t, x, u] = ∑[−ws(xs) − ((∇vs(xs))T (∑ Rsj(t, x) + Bs(xs)us))+
s=1 j According to Rumyantsev and Krasovsky theorems optimal control and optimal Lyapunov function we obtain from a system + 21 usT βs(xs)us)] + Ψ1(t, x).
u0 = (u1T 0 , ..., usT 0 )T , ∂B ∂us
= Bs(xs) + βs(xs)us0 = 0, s = 1, q.
us0 = −βs 1(xs)(∇vs)T Bs(xs) = −βs 1(xs)BsT (xs)∇vs.
From (4) we have Substituting us0 in (5) to function B[v; t, x, u] we have the algebraic equation B[v0; t, x, u0] = 0 with respect to function Ψ1(t, x).
The function
q q Ψ1 = − ∑ ws(xs) − ((∇vs(xs))T (∑ Rsj(t, x))) + (usT )0βs(xs)us0,
s=1 j=1 will be positive definite and functional of control finally becomes
J (u0) = + (us0)T βs(xs)us0 + usT βs(xs)us]dt.
We applied to system (1) and to functional (7) the results about optimal stabilization of common nonhomogeneous multiply connected systems [Shchennikova, 2006]. In the case under consideration we use general scheme of stabilization without assumption of positive definition of (6) because this function has required property. The algorithms of optimal stabilization of system (1) with respect to all and to a part of variables are developed. 3
The Optimal Stabilization with Respect to a Part of Variables We consider multiply connected nonlinear controlled dynamic system dxs = fs(t, xs, ulsoc) + Fs(t, x, usglob) ≡ Φs(t, x, ulsoc, usglob), s = 1, q, dt where x = (x1T , ..., xqT )T , xs ∈ Rns , Rn1 ⊕ ... ⊕ Rnq = Rn, ulsoc(t, 0) = 0, usglob(t, 0) = 0, Φs(t, 0, 0, 0) ≡ 0.
It is accepted that right part of system (8) is defined in domain and conditions of existence and uniqueness of solution are satisfied. Let us assume than system (8) can be represented as
Ω1 = {t, x, ulsoc, usglob : t ≥ t0 ≥ 0, ||x|| < H, ||ulsoc|| < ∞, ||usglob|| < ∞, 0 < H = const, s = 1, q},
q dys = Ys(t, ys, zs, ulsoc) + ∑ Y1sj(t, y, z)usglob, dt j=1
q dzs = Zs(t, ys, zs, ulsoc) + ∑ Z1sj(t, y, z)usglob, dt j=1 (4) (5) (6) (7) (8) (9) (10) where xs = (ysT , zsT )T , x = (yT , zT )T , where ys ∈ Rks , zs ∈ Rms , ks + ms = ns, s = 1, q. For system (10) domain (9) takes the form Ω2 = {t, x, ulsoc, usglob : t ≥ t0 ≥ 0, ||ys|| < Hs, ||zs|| ≤ ∞,
||ulsoc|| < ∞, ||usglob|| < ∞, 0 < H = const, s = 1, q}, and each solution is z-extendible.
We consider the subsystems of the form dys = Ys(t, ys, zs, ulsoc) dt dzs = Zs(t, ys, zs, ulsoc), s = 1, q.
dt
Further, we will solve the problem of optimal ys-stabilization of multiply connected dynamical systems of the form (10), s = 1, q, y = (y1T , ..., yqT )T , using the method of Lyapunov vector-functions. In this case the strategy of solving the problem of stabilization is that each subsystem must be ys-stabilized with the help of local controls ulsoc, s = 1, q, i.e. it must be ys-stabilized on the level of subsystems, and then the asymptotic ys-stability of interconnected subsystems must be checked. The general scheme of a two-level stabilization scheme [Shil’yak, 1994] is that the global control uglob, s = 1, q, is added to the decentralized control in order to weaken s the effect of interrelated subsystems. In this work the problem of optimal stabilization of multiply connected system is sold also using a two-level stabilization scheme with respect to a part of variables.
We consider the case when right parts of (11) can be written in the form
Ys(t, ys, zs, ulsoc) ≡ Y s(t, ys, zs) + b1s(t, ys, zs)ulsoc1,
Zs(t, ys, zs, ulsoc) ≡ Zs(t, ys, zs) + b2s(t, ys, zs)ulsoc2, s = 1, q, where b1s(t, ys, zs) and b2s(t, ys, zs) are matrixes of appropriate dimensions, and controls uloc1 and ulsoc2 are built s considering the choice of Lyapunov vector functions.
It was shown that equilibrium state of system (11) taking into account (12) is uniformly asymptotic ys-stable. In this case system (10) can be represented in the form (11) (12) (13) (14) where
We consider the problem of optimal stabilization for system (13). Criterion of quality control we write in the integral form in this case we define the function w(t, x, u) in the process of solving.
As optimal Lyapunov function for system (13) we choose the function dys = φs(t, ys, xs) + Y1s(t, ye, ze)usglob, dt dzs = ψs(t, ys, zs) + Z1s(t, ye, ze)usglob, dt φs(t, ys, zs) = Y (t, ys, zs) + b1s(t, ys, zs)ulsoc(t, ys, zs), ψs(t, ys, zs) = Zs(t, ys, zs) + b2s(t, ys, zs)ulsoc(t, ys, zs).
1 ∫ 0 J =
w(t, y[t], z[t], usglob[t])dt, V (t, y, z) =
q ∑ αsVs(t, ys, zs), s=1 where αs are positive real constants, Vs(t, xs) are Lyapunov functions which guarantee uniform asymptotic ys-stability of systems (11), s = 1, q.
We introduce Krasovsky–Bellman function B(t, x, u, v, uglob) with special component Ψ(t, y, z) allowing to consider the function w in integral (14) in the form q w(t, y, z, uglob) = Ψ(t, y, z, uglob) + 1 ∑(uglob)T θsusglob.
2
s=1
According to Rumyantsev theorem function B(t, x, u, v, uglob) is positive-definite with respect to y. Along optimal control (us0)glob we have that
In result we obtain positive-definite function with respect to y-component of phase vector of system (10). In this case we can write the criterion of quality control in the form ∫1( ∑q αsWs(t, ys, zs) + ∑q θsj (us0)glob(uj0)glob + ∑q θsj usglobujglob)dt.
Acknowledgements This work was supported by the Presidium of the Russian Academy of Sciences, through program no. I.31, Challenging Problems of Robotics.